Constraining Palatini-Horndeski theory with gravitational waves after GW170817

In this paper, we investigate the possible parameter space of Palatini-Horndeski theory with gravitational waves in a spatially flat Universe. We develop a general method for obtaining the speed of gravitational waves in the Palatini formalism in the cosmological background and we find that if the theory satisfies the following condition: in any spatially flat cosmological background, the tensor gravitational wave speed is the speed of light c, then only S = $\int$ d${}^4$x $\sqrt{-g}$ [K($\varphi$, X)-G${}_3$($\varphi$, X)$\widetilde{◻}\varphi$ + G${}_4$($\varphi$)$\tilde{R}$] is left as the possible action in Palatini-Horndeski theory. We also find that when G${}_5$($\varphi$, X) $\ne$ 0, the tensor part of the connection will propagate and there are two different tensor gravitational wave speeds.